# What is Statistic: Definition and 122 Discussions

A statistic (singular) or sample statistic is any quantity computed from values in a sample that is used for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypothesis. The average (aka mean) of sample values is a statistic. The term statistic is used both for the function and for the value of the function on a given sample. When a statistic is being used for a specific purpose, it may be referred to by a name indicating its purpose.
When a statistic is used to estimate a population parameter, the statistic is called an estimator. A population parameter is any characteristic of a population under study, but when it is not feasible to directly measure the value of a population parameter, statistical methods are used to infer the likely value of the parameter on the basis of a statistic computed from a sample taken from the population. For example, the mean of a sample is an unbiased estimator of the population mean. This means that the expected value of the sample mean equals the true mean of the population.In descriptive statistics, a descriptive statistic is used to describe the sample data in some useful way. In statistical hypothesis testing, a test statistic is used to test a hypothesis. Note that a single statistic can be used for multiple purposes – for example the sample mean can be used to estimate the population mean, to describe a sample data set, or to test a hypothesis.

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1. ### I Definition of Uncertainity

I have some confusion regarding Measurement Uncertainity. In some books/articles it is defined wrt true value as "Uncertainty in the average of measurements is the range in which true value is most likely to fall , when there is no bias or systematic component of error is involved in...
2. ### I Probability to get a certain number or less by throwing two 6-faced die

In a board game, I need to reach a certain place in a board divided into boxes, I move by throwing 2 six-faced dices. If my goal is 4 boxes away I need to obtain at least a combination of numbers that sums to 4, but any higher number is also a favorable outcome. I want to calculate the...
3. ### B Concerning the birthday problem in probability

The problem is stated like this : There are k people in a room. Assume each person’s birthday is equally likely to be any of the 365 days of the year (we exclude February 29), and that people’s birthdays are independent (we assume there are no twins in the room). What is the probability that two...
4. ### A Finding Minimal Mean Distance Curves on the Unit Sphere

**Problem:** Find parametric equations for a simple closed curve of length 4π on the unit sphere which minimizes the mean spherical distance from the curve to the sphere; the solution must include proof of minimization. Can you solve this problem with arbitrary L > 2π instead of 4π? There...
5. ### B Where is the error in my reasoning about palindromes?

Hello everyone, I found this problem online about probability, for me, I think that to have a 2 letter palindrome is less likely because we need to have the same letter in the 2 places which gives us 26 possibilities (aa , bb, cc ....) however for words with 3 letters we have 26 possibilities...
6. ### B About the naive definition of probability

hello, I took an introductory course about statistics, we viewed the naive definition of probability which says "it requires equally likely outcomes and can't handle an infinite sample space ", I understood that it requires finite sample space but I didn't understand "equally likely outcomes "...
7. ### Probability of Hypokalemia w/ 1 or Multiple Measurements

TL;DR Summary: Finding the probability with one measurement and multiple measurements on separate days. Question: Hypokalemia is diagnosed when blood potassium levels are low, below 3.5 mmol/L. Let’s assume we know a patient whose measured potassium levels vary daily according to N(µ = 3.8...
8. ### I Shapiro-Wilks test and the order statistic

Given the Shapiro-Wilk test value W: where I'm interested in the numerator. If my data is sorted in ascending order, my understanding is that $x_(i) = x_i$. Is that correct?
9. ### Finding all possible energy states faster without using a calculator

I tried to find states in direct method using ##\frac{E}{E_0}=\:nx^2+ny^2+nz^2## and ##100\:<nx^2+ny^2+nz^2\:<\:136## But it was too long, found it using phi approximation there are around 300 energy states, and Python find around 271 states using direct method but I need manual or recursive...

11. ### A Calculating the variance of integrated Poisson noise on a defined quantity

It is in cosmology context but actually, but it is also a mathematics/statistical issue. From spherical harmonics with Legendre deccomposition, I have the following definition of the standard deviation of a ##C_\ell## noised with a Poisson Noise ##N_p## : ## ...
12. ### MHB Give Possible Statistic

Give a possible statistic about each item: a. a football team b. a fourth-grade class c. one of your classes
13. ### A Computing a variance in astrophysics context

Below the error on photometric galaxy clustering under the form of covariance : $$\Delta C_{i j}^{A B}(\ell)=\sqrt{\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell}}\left[C_{i j}^{A B}(\ell)+N_{i j}^{A B}(\ell)\right]$$ where ##_{\text {sky }}## is the fraction of surveyed sky and ##A, B##...
14. ### Bayes' theorem and disease prevalence

Hello at all! I have to solve this exercise: A tampon diagnostic test provides 1% positive results. The positive predictive values (probabilities of positive test disease) and negative (absence disease given negative test) are respectively 0.95 and 0.98. What is the prevalence of the disease...
15. ### I Statistic uncertainties in cross section plots - how to calculate?

Hi there, I hope I chose the right forum for my question. So, basically, I'm doing an analysis measuring the number of signal particles in a certain momentum bin i, and doing two corrections: Nsig, i=M*(Nmeas, i-Nbkg, i) Here, M is a matrix covering PID correction and PID efficiencies, and...
16. ### LOGPEARSONDIST and LOGNORMALDIST Function in PHP

for reference you can see JS in formulajs.info/functions. lognormdist use by call : formulajs.LOGNORMDIST(value, mean, stdev, true) logpearsondist use by call : formulajs.NORMSDIST(z, true) anybody can help?
17. ### Sum of the Expected Values of Two Discrete Random Variables

Apologies if this isn't the right forum for this. In my stats homework we have to prove that the expected value of aX and bY is aE[X]+bE[Y] where X and Y are random variables and a and b are constants. I have come across this proof but I'm a little rusty with summations. How is the jump from the...
18. ### I Conditional distribution of geometric series

Can someone help me on this question? I'm finding a very strange probability distribution. Question: Suppose that x_1 and x_2 are independent with x_1 ~ geometric(p) and x_2 ~ geometric (1-p). That's x_1 has geometric distribution with parameter p and x_2 has geometric distribution with...
19. ### I Nonparametric Hypothesis Tests

Hello everyone, Say you have two sample distributions that are known to be two different distributions (one randomly drawn from a Poisson distribution, other randomly drawn from an uniform distribution). Given that you know the distributions are not going to be normal, a two-sample t-test would...
20. ### Size 6: Is It Right for Me?

is the most common size 6 , is my taking right, if not please provide explanation please
21. ### I Statistics proof: y = k x holds for a data set

Simple linear regression statistics: If I have a linear relation (or wish to prove such a relation): y = k x where k = constant. I have a set of n experimental data points ...(y0, x0), (y1, x1)... measured with some error estimates. Is there some way to present how well the n data points shows...
22. ### Two teams, A and B, are playing a series of games

My attempt I used negative binomial to solve the problem, however I'm left with a polynomial that is difficult to solve? Is there any other way to approach this problem? I used the inequality because I'm trying to find the range of p. Since the probability of winning the series for team...
23. ### I Discrete to continuum Gaussian function

I have a question regarding a paragraph in "Radiation detection and measurement" by Knoll. In the chapter about the discrete Gaussian it states that "Because the mean value of the distribution ##\bar{x}## is large , values of ##P(x)## for adjacent values of x are not greatly different from each...
24. ### I Uncertainty of MonteCarlo Simulations: Weight and Error Bars

Hello everybody, I need a help, primarly a confirmation about my reasoning. I have data from a MonteCarlo simulation of collisions between particles at LHC (made with Madgraph). I have plotted some variables, for example the angle between two final leptons. Then I have normalized the plot to a...
25. ### A Estimation error from estimation quantile of normal distribution

Hi guys, For my (master) project I am trying to find the probability that a random variable, which is normally distributed, exceeds a quantile that is estimated by a limited number of observations. See attached for my attempt. - Is it correct? - How to incorporate the fact that the mean and...
26. ### A Similarity between -/+ weighted distributions

Suppose that you have +/- elements aggregated into a weighted distribution function that represents some deviation from an unknown background distribution. What would be a good similarity metric for comparing two such distributions (2D or 3D), if they each represent different perturbations...
27. ### A Defining a Symmetry Statistic to test for Normality?

Hi, Just curious. One of the requirements for a data set to be normally-distributed is that of symmetry. Question: Is there or could one define a statistic to this effect, meaning to determine the level of symmetry on the data set and use that to determine at certain confidence level if the...
28. ### I Small Reduced Chi Squared interpretation

Hello everyone, I would be happy if someone explained the small reduced chi squared value to me. I have fitted a set of measured data with an exponential function, which I need for some sw calculations. The fit seams great, the origin sw also provides the reduced chi squared, but it is very...
29. ### Bayesian Probability Distributions

Hi, I was having some trouble doing some bayesian probability problems and was wondering if I could get any help. I think I was able to get the first two but am confused on the last. If someone could please check my work to make sure I am correct and help me on the last question that would be...
30. ### I R-squared statistic for goodness-of-fit

Hello all, While I was reading about linear regression, I stumbled on the concept r-squared statistic that measures the goodness-of-fit of the line to the data points. It is defined as: R^2 = 1 - \frac{\sum_i (y_i - f(x_i))^2}{\sum_i (y_i - E[y])^2} where f(x_i) is the fitted/predicted...
31. ### Conceptual Question regarding hypothesis testing regression

Homework Statement Hi, I had a question regarding testing a regression models coefficients. Say there is a regression model that has the form: y = b0 + b1x1 + b2x2 + b3x3 + b4x4 + e For the sake of simplicity let: e be the random error, x1 is age, x2 is severity, and x3 is anxiety. y is...
32. L

### I Problems that could occur in estimating n from a Binomial distribution

Hi, I am doing the following question: https://i.gyazo.com/f2e651334bcbd5f1dcb6d661e4160956.png I have estimated both n and theta. But the part that is throwing me off is what problem could you encounter in estimating n here? My only idea is that it might be something to do with the sample...
33. D

### I How does the distribution depend on a variable resolution

Dear all, We were trying to solve the following question but did not quite understand what to do. The question is as follows: The reconstructed invariant mass is usually described by a Gaussian (or Normal) distribution. However, the resolution σ (the width of the distribution) is found to...
34. G

### Minimize the sum of Type I and Type II errors

Homework Statement Given X_1,\dots,X_n a simple random sample with normal variables (\mu, \sigma^2). We assume \mu is known but \sigma^2 is unknown. The hypothesis is \begin{cases} H_0: & \mu=\mu_0 \\ H_1: & \mu=\mu_1 > \mu_0 \end{cases} Determine the rejection region R...
35. ### Confusion between z and t values

Homework Statement Hi, Alright so I have some confusion on when to use specific tests and the z vs t test. Given this example (not my homework) could someone please clarify. Alright say you have a random sample of size 200. You find the sample mean to be 10 and the sample standard deviation...
36. ### Help on determining independent events

Homework Statement Hi, I have this question that I've been pondering for a while, I keep flipflopping on what I think is right. I only need help on the last part on whether the events are independent or not, the rest of the text is backstory to the question. I know for events to be independent...
37. ### Probability theory and statistics for Robotics and ME

I study control theory and robotics. Recently I figured out that I have a much deeper understanding of probability and statistics compared to my colleagues. Is this 'talent' valuable in my field and if so, where? We used this theory to define white noise, but nothing more...as of now. Also I am...
38. ### Chi-square goodness of fit cannot find expected values

Homework Statement An article in Business Week reports profits and losses of firms by industry. A random sample of 100 firms is selected, and for each firm in the sample, we record whether the company made money or lost money, and whether or not the firm is a service company. The data are...
39. ### I Calculating Probability with Poisson Distribution: Radioactive Source Counts

Hello! I came across this problem: A counter near a long-lived radioactive source measures an average of 100 counts per minute. What is the probability that more than 110 counts will be recorded in a given one-minute interval? I am not sure how to do it. Applying the Poisson distribution formula...
40. ### I What is Yates' correction of contingency?

I can't understand a word of https://en.wikipedia.org/wiki/Yates's_correction_for_continuity']Wikipedia.[/PLAIN] [Broken] P.S. What I know so far is subtracting 0.5 from O-E if df=1 in Goodness of fit (?) is Yates' correction.
41. ### I Level of significance and acceptance and rejection of the null hypothesis

Why do we reject the null hypothesis in Goodness of fit when the Chi square statistic is less than the tabulated value of chi-square at say 5% level of significance and accept when it is more? What does it mean to have a Chi-square value more or less than the value assigned to a certain level...
42. ### I Understanding the null hypothesis

I was reading Bio-statistics principles and practice by Antonisamy and stumbled upon the following: Null hypothesis is a hypothesis that suggests an absence of difference, association or effect, the negation of which provides evidence for presence of difference, association or effect. The only...
43. ### Maximum likelihood of a statistical model

Homework Statement I look at the distribution ##(Y_1,Y_2,...,Y_n)## where ##Y_i=μ+(1+φ x_i)+ε_i## where ##-1<φ<1## and ##-1<x_i<1## . x's are known numbers. ε's are independent and normally distributed with mean 0 and variance 1. I need to find the the maximum likelihood estimator for μ and...
44. ### Calculating Energy Levels in a Ni2+ Ion Crystal

Hello, I am stuck at the beginning of an exercise because I have some trouble to understand how are the energy level in this problem : In a crystal we have Ni2+ ions that we consider independent and they are submitted to an axial symmetry potential. Each ion acts as a free spin S=1. We have the...
45. ### A What is first and Second order Dependence?

Can someone please explain to me what it means what they say a model is "first and second order dependence?"
46. ### Busy Barber Problem: Proportion of Time Apprentice is Busy

Homework Statement A barbershop has two barbers: an experienced owner and an apprentice. The owner cuts hair at the rate of 4 customers/hour, while the apprentice can only do 2 customers/hour. The owner and the apprentice work simultaneously, however any new customer will always go first to the...
47. ### Long-run proportion in state ′A′

The question asks: A physical device can be in three states: A,B,C. The device operates as follows (all time units are in hours): The device spends an exponentially distributed amount of time in stateAA (with mean of 12minutes) and then with probability 0.6 goes to state B, and with...
48. ### I Probability of Particle Collision on a Closed Surface

Is it more probable that particles will collide if both are moving or if one has velocity equal to 0? Let's say we don't have any forces between them and they're on a closed surface (for example a square).
49. ### Roll a fair die 5 times

Homework Statement Roll a fair die 5 times, find the probability that the first two rolls have the same outcomes. Homework EquationsThe Attempt at a Solution The total outcomes is 6^5, I think we have 6^2 * 6 choose 2 /6^5 since the first two numbers are fixed and we can choose 2 numbers from...
50. ### I Proof to the Expression of Poisson Distribution

Hello. Given a range of time in which an event can occur an indefinite number of times, we say a random variable X folows a poisson distribution when it follows this statements: X is the number of times an event occurs in an interval and X can take values 0, 1, 2, … The occurrence of one event...